[proofplan]
The positivity of $L$ gives, after replacing $L$ by a positive tensor power, a holomorphic embedding of $X$ into projective space whose hyperplane bundle restricts to that tensor power of $L$. Projective Serre vanishing then proves the desired cohomology vanishing for the subsequence of powers divisible by that tensor power. The finitely many remaining congruence classes are handled by applying the same divisible-power argument to the coherent sheaves $\mathcal F\otimes L^a$ for $0\le a<r$ and then taking a maximum of finitely many thresholds.
[/proofplan]
[step:Embed $X$ by a positive tensor power of $L$]
By the [Kodaira embedding theorem](/theorems/3836) for positive holomorphic line bundles, applied to the positive line bundle $L\to X$, there exist integers $r\ge 1$ and $N\ge 0$ and a closed holomorphic embedding
\begin{align*}
i:X\longrightarrow \mathbb P^N
\end{align*}
such that
\begin{align*}
i^*\mathcal O_{\mathbb P^N}(1)\cong L^r.
\end{align*}
Here $\mathcal O_{\mathbb P^N}(1)$ denotes the hyperplane line bundle on $\mathbb P^N$. We use this isomorphism throughout to identify
\begin{align*}
i^*\mathcal O_{\mathbb P^N}(k)\cong L^{rk}
\end{align*}
for every integer $k\ge 0$.
The [Kodaira embedding theorem](/theorems/3844) is used here as a standard prerequisite not yet resolved to a wiki theorem citation: [Kodaira embedding theorem](/theorems/3871).
[/step]
[step:Translate projective Serre vanishing to powers divisible by $r$]
Let $\mathcal G:=i_*\mathcal F$ be the direct image sheaf on $\mathbb P^N$. Since $i:X\to\mathbb P^N$ is a closed holomorphic embedding and $\mathcal F$ is coherent on $X$, the sheaf $\mathcal G$ is coherent on $\mathbb P^N$. By the projection formula for a closed embedding, for every integer $k\ge 0$ there is a natural isomorphism
\begin{align*}
\mathcal G\otimes_{\mathcal O_{\mathbb P^N}}\mathcal O_{\mathbb P^N}(k)\cong i_*(\mathcal F\otimes_{\mathcal O_X} i^*\mathcal O_{\mathbb P^N}(k)).
\end{align*}
Using $i^*\mathcal O_{\mathbb P^N}(k)\cong L^{rk}$, this becomes
\begin{align*}
\mathcal G(k)\cong i_*(\mathcal F\otimes L^{rk}),
\end{align*}
where $\mathcal G(k):=\mathcal G\otimes_{\mathcal O_{\mathbb P^N}}\mathcal O_{\mathbb P^N}(k)$.
Since $i$ is a closed embedding, the direct image functor $i_*$ is exact and cohomology is unchanged by viewing a sheaf on $X$ as its direct image on $\mathbb P^N$. Hence, for every integer $q\ge 0$ and every integer $k\ge 0$,
\begin{align*}
H^q(\mathbb P^N,\mathcal G(k))\cong H^q(X,\mathcal F\otimes L^{rk}).
\end{align*}
Projective Serre vanishing, applied to the coherent sheaf $\mathcal G$ on $\mathbb P^N$, gives an integer $k_0\in\mathbb N$ such that
\begin{align*}
H^q(\mathbb P^N,\mathcal G(k))=0
\end{align*}
for every $q>0$ and every $k\ge k_0$. Therefore
\begin{align*}
H^q(X,\mathcal F\otimes L^{rk})=0
\end{align*}
for every $q>0$ and every $k\ge k_0$.
Projective Serre vanishing, the projection formula for closed embeddings, and the cohomology invariance under a closed embedding are used here as standard prerequisites not yet resolved to wiki theorem citations.
[guided]
The embedding lets us replace the analytic statement on $X$ with a projective-space statement. Define
\begin{align*}
\mathcal G:=i_*\mathcal F.
\end{align*}
Because $i:X\to\mathbb P^N$ is a closed holomorphic embedding and $\mathcal F$ is coherent, the direct image $\mathcal G$ is a coherent sheaf on $\mathbb P^N$.
Now fix an integer $k\ge 0$. The projection formula for the closed embedding $i$ gives
\begin{align*}
i_*\mathcal F\otimes_{\mathcal O_{\mathbb P^N}}\mathcal O_{\mathbb P^N}(k)\cong i_*(\mathcal F\otimes_{\mathcal O_X} i^*\mathcal O_{\mathbb P^N}(k)).
\end{align*}
The point of choosing the embedding from the positivity of $L$ is exactly that
\begin{align*}
i^*\mathcal O_{\mathbb P^N}(1)\cong L^r.
\end{align*}
Taking the $k$-th tensor power gives
\begin{align*}
i^*\mathcal O_{\mathbb P^N}(k)\cong L^{rk}.
\end{align*}
Substituting this into the projection formula yields
\begin{align*}
\mathcal G(k)\cong i_*(\mathcal F\otimes L^{rk}).
\end{align*}
We next compare cohomology. Since $i$ is a closed embedding, the direct image functor $i_*$ is exact, and sheaf cohomology of a sheaf on $X$ agrees with the cohomology of its direct image on $\mathbb P^N$. Therefore, for every integer $q\ge 0$,
\begin{align*}
H^q(\mathbb P^N,\mathcal G(k))\cong H^q(X,\mathcal F\otimes L^{rk}).
\end{align*}
Projective Serre vanishing says that for the coherent sheaf $\mathcal G$ on projective space, there exists an integer $k_0\in\mathbb N$ such that
\begin{align*}
H^q(\mathbb P^N,\mathcal G(k))=0
\end{align*}
for all $q>0$ and all $k\ge k_0$. Combining this with the cohomology comparison gives
\begin{align*}
H^q(X,\mathcal F\otimes L^{rk})=0
\end{align*}
for all $q>0$ and all $k\ge k_0$. This proves the theorem for the powers of $L$ whose exponent is divisible by $r$.
[/guided]
[/step]
[step:Handle each residue class modulo $r$]
For each integer $a$ with $0\le a<r$, define the coherent sheaf
\begin{align*}
\mathcal F_a:=\mathcal F\otimes L^a.
\end{align*}
The [tensor product](/page/Tensor%20Product) of a coherent sheaf with a holomorphic line bundle is coherent, so each $\mathcal F_a$ is coherent on $X$.
Applying the previous step to $\mathcal F_a$ in place of $\mathcal F$, there exists an integer $k_a\in\mathbb N$ such that
\begin{align*}
H^q(X,\mathcal F_a\otimes L^{rk})=0
\end{align*}
for every $q>0$ and every $k\ge k_a$. Since
\begin{align*}
\mathcal F_a\otimes L^{rk}\cong \mathcal F\otimes L^{a+rk},
\end{align*}
we obtain
\begin{align*}
H^q(X,\mathcal F\otimes L^{a+rk})=0
\end{align*}
for every $q>0$ and every $k\ge k_a$.
[/step]
[step:Choose one threshold that works for all powers]
Define the integer
\begin{align*}
m_0:=\max_{0\le a<r}(a+rk_a).
\end{align*}
Let $m\ge m_0$ be an integer. By Euclidean division, there exist unique integers $a$ and $k$ such that
\begin{align*}
0\le a<r
\end{align*}
and
\begin{align*}
m=a+rk.
\end{align*}
Since $m\ge m_0\ge a+rk_a$, it follows that $k\ge k_a$. The vanishing obtained for the residue class $a$ gives, for every $q>0$,
\begin{align*}
H^q(X,\mathcal F\otimes L^m)=H^q(X,\mathcal F\otimes L^{a+rk})=0.
\end{align*}
Thus the same integer $m_0$ works for every $q>0$ and every $m\ge m_0$, which proves the theorem.
[/step]
